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Groups with various types of operators, in particular the recently introducedRota-Baxter groups, have generated renowned interest with close connections tonumerical integrals, Yang-Baxter equation, integrable systems and post-Hopfalgebras. This paper gives the general notion of operated groups and providesexplicit constructions of free operated groups, free differential groups andfree Rota-Baxter groups.

Operated groups, differential groups and Rota-Baxter groups with an emphasis on the free objects